All values outside this interval are potential outliers, or might suggest that the normality assumption is wrong. These residuals, given the assumptions of the linear regression model, should be normally distributed, meaning that 95% of the residuals should be in the interval. It enables us to take a closer look at each of the standardized residuals. If we look at the parameter corresponding to Gender-f, it seems that for a given age and height, being a girl means a small inccrease of the weight. This confirms the weak impact of the Gender on the model. We can see that the p-value for the Gender parameter is 0.83, and that the corresponding confidence range includes 0. This table is helpful when predictions are needed, or when you need to compare the coefficients of the model for a given population with the ones obtained for another population. The following table gives details on the model. We can see that the gender brings the least information to the model. The lower the F probability corresponding to a given variable, the stronger the impact of the variable on the model. The Type III SS is generally the best method to use to interpret results when an interaction is part of the model. This means that the order in which the variables are selected will not have any effect on the values in the Type III SS. The Type III SS table is computed by removing one variable of the model at a time to evaluate its impact on the quality of the model. We can see here that the Gender bring only little information to the model, once the Height and the Age have been added. The lower the F probability corresponding to a given variable, the stronger the impact of the variable on the model as it is before the variable is 3Ĥ added to it. In consequence, in Type I SS, the order in which the variables are selected will influence the results. The Type I SS table is constructed by adding variables in the model one by one, and by evaluating the impact of each on the model sum of squares (Model SS). To do this, we have to examine the Type I SS and Type III SS tables (see below). We also want to find out if the three variables provide the same amount of information. Therefore, we can conclude with confidence that the three variables do bring a significant amount of information. Given the fact that the probability corresponding to the F value is lower than, it means that we would be taking a lower than 0.01% risk in assuming that the null hypothesis (no effect of the two explanatory variables) is wrong. In other words, it's a way of asking yourself whether it is valid to use the mean to describe the whole population, or whether the information brought by the explanatory variables is of value or not. The results enable us to determine whether or not the explanatory variables bring significant information (null hypothesis H0) to the model. It is important to examine the results of the analysis of variance table (see below). We can guess that some genetic and nutritive effects are involved, but it might be that simply by transforming the available variables we could obtain some better results. The remainder of the variability is due to some effects (other explanatory variables) that have not been or that could not be measured during this experiment. 1ģ In this particular case, 63 % of the variability of the Weight is explained by the Height, the Age and the Gender. Setting up an ANCOVA After opening XLSTAT, select the XLSTAT / Modeling data / ANCOVA command, or click on the corresponding button of the Modeling Data toolbar (see below). In two other tutorials on linear regression this dataset is also used, with the Height and then the Height and the Age as explanatory variables. The specificity of ANCOVA is that it mixes qualitative and quantitative explanatory variables. The ANCOVA method belongs to a larger family of models called GLM (Generalized Linear Models) as do the linear regression and the ANOVA. Goal of this Analysis of Covariance (ANCOVA) Using the Analysis of Covariance (ANCOVA), we want to find out how the weight of the children varies with their gender (a qualitative variable that takes value f or m), their height and their age, and to verify if a linear model makes sense. They concern 237 children, described by their Gender, Age in months, Height in inches (1 inch = 2.54 cm), and Weight in pounds (1 pound = 0.45 kg). Introduction to Experimental Ecology, New York: Academic Press, Inc. 1 Running an ANCOVA in XLSTAT demoanco.xls Dataset for running an ANCOVA An Excel sheet with both the data and results used in this tutorial can be downloaded by clicking here.
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